Having defined both \(\mathbf {\zeta }_{0}^{x,x}\)  and \(\mathbf {\zeta }_{2}^{x,x}\) , the third–order susceptibility follows from [1]} (with \(\omega _{\sigma }=\omega _{1}+\omega _{2}+\omega _{3}\) ) as
\(\chi _{x,x,x,x}^{\left(3\right)}\left(-\omega _{\sigma };\omega _{1},\omega _{2},\omega _{3}\right) & =\frac{1}{3 !} \mathcal {P}\left\lbrace -\mathbf {c}^{\dagger }\left(\left(-\omega _{\sigma }\right)^{\dagger }\right)\cdot \left[\mathbb {T}_{0} \cdot \zeta _{0}^{x,x}\left(-\omega _{2},-\omega _{3}\right)+\mathbb {T}_{2} \cdot \zeta _{2}^{x,x}\left(-\omega _{2},-\omega _{3}\right)\right]\right.+ \nonumber \\& \left.+\left[ \mathbf {S} \cdot \mathbf {c}\left(-\omega _{\sigma }\right) \right] \times \left[\sum _{n=1}^{N}c_{n}^{\dagger }\left(\left(-\omega _{2}\right)^{\dagger }\right)c_{n}\left(\omega _{1}\right)2\pi \frac{R^{2}}{2}\left[J_{2}\left(\mathrm {z}_{1, n}\right)\right]^{2}\right]\right\rbrace ,\) 
